Problem 5 - Elementary Algebra

Question

\(\left(32 \mathrm{~mm}^{2}\right)(4 \mathrm{~mm})\)

Step-by-Step Solution

Answer

128 mm³

Step 1: Understand the Problem

The exercise asks to multiply two values given in terms of millimeters: 32 mm² and 4 mm.

Step 2: Multiply the Numbers

Multiply the numerical values from the two terms. This results in: 32 * 4 = 128.

Step 3: Combine the Units

When multiplying units of area (mm²) with a unit of length (mm), the resulting unit will be in terms of volume (mm³). Thus, the unit becomes mm² * mm = mm³.

Step 4: Write the Final Answer

Combining both the numerical result and the units, we obtain: 128 mm³.

Key Concepts

Unit Conversion

Unit conversion plays a critical role in ensuring that measurements are accurate and meaningful. Units must be compatible when performing operations like addition, subtraction, multiplication, or division.

In the given exercise, we need to understand how units change when they are multiplied.

Here, we start with 32 mm² (square millimeters) and 4 mm (millimeters). The key is to recognize how units behave mathematically:

• When you multiply a unit of area (like mm²) by a unit of length (like mm), the resulting unit represents a volume (like mm³).
  Therefore, converting and understanding units can help you grasp how different measurements relate.

Multiplication of Measurements

Multiplying measurements involves not just the numerical values but also their respective units.

Let's follow the steps:

• First, multiply the numerical values:
  32 * 4 = 128.
• Next, combine the units:
  mm² * mm = mm³.


The correct method of multiplying units ensures the final answer accurately represents the physical quantity being measured.

Another example to illustrate this is: if you multiply 5 cm (centimeters) by 2 cm² (square centimeters), the result will be 10 cm³ (cubic centimeters).

Algebraic Operations

Algebraic operations are fundamental in solving mathematical problems. They guide you in manipulating both numbers and variables systematically.

In our example, we perform two main algebraic operations: multiplication of numbers and multiplication of units.

• Multiplying Numbers:
  This is straightforward. Simply multiply the numerical parts: 32 * 4.
• Multiplying Units:
  Apply the rules of exponent addition (when the bases are the same). Here, mm² * mm is treated as mm²+¹, resulting in mm³.


Understanding these core algebraic principles helps simplify complex problems.

In algebra, always remember to follow order of operations and keep track of unit conversion to get accurate results.